Abstract

We study the efficiency of a simple quantum dot heat engine at maximum power.
In contrast to the quasi-statically operated Carnot engine whose efficiency reaches the theoretical maximum,
recent research on more realistic engines operated in a finite time has revealed other classes of efficiencies
such as the Curzon-Ahlborn efficiency maximizing the power. Such a power-maximizing efficiency has been argued
to be always the half of the maximum efficiency up to the linear order near equilibrium under the tight-coupling condition between thermodynamic fluxes.
We show, however, that this universality may break down for the quantum dot heat engine,
depending on the constraint imposed on the engine control parameters, even though the tight-coupling condition remains satisfied.
It is shown that this deviation is critically related to the applicability of the linear irreversible thermodynamics.

In this paper, we take a quantum dot heat engine composed of a single quantum dot connected to two leads with characteristic temperatures and chemical potentials Esposito2009EPL (); Esposito2012 (); Toral2016 (); exp () to elucidate the condition for the maximum power (Fig. 1).
We consider various restricted control-parameter spaces to maximize the power output and find an intriguing result: When the quantum dot energy level relative to one of the lead’s chemical potential is fixed and the other is varied, the linear coefficient of the power-maximizing efficiency ηop near equilibrium takes the conventional CA value of 1/2, i.e. ηop≈12ηC for small ηC (Carnot efficiency) VanDenBroeck2005 (), but
its quadratic coefficient deviates from the CA value of 1/8, which has been already noticed in a previous study Esposito2009PRL ().
On the other hand, when the quantum dot energy level is varied with fixed chemical potentials of both leads, we find that
even the linear coefficient deviates from 1/2 that has been believed to be “universal” for any tight-coupling engine VanDenBroeck2005 (); Esposito2009PRL (). In this case, the linear coefficient turns out to be unity (ηop≈ηC), which implies a much higher
efficiency at maximum power, compared to the conventional cases.

Figure 1: A schematic illustration of the quantum dot heat engine. The quantum dot with a single energy level EG is in contact with the leads, which plays the role of heat and particle reservoirs with temperatures T1 and T2, and chemical potentials μ1 and μ2. The maximum (Carnot) efficiency is given as ηC=1−T2/T1.

We emphasize that our engine always satisfies the tight-coupling condition in the sense that the heat flux is directly
proportional to the work-generating flux VanDenBroeck2005 (); Esposito2009PRL (). This implies that the universality
requires an additional constraint besides the tight-coupling condition, which turns out to be the applicability of the linear
irreversible thermodynamics Groot (). We point out that the latter non-universal case is also experimentally realizable as it corresponds to tuning the gate voltage of the quantum dot to optimize the power exp (), while the control of the chemical potential difference of the leads can be done by adjusting the source-drain voltage Kouwenhoven1997 (); YSLiu2013 (); Humphrey2002 (); Jordan2013 (). In a recent experiment controlling the gate voltage, much higher efficiency than the usual CA efficiency was reported at maximum power exp (), which supports our result.

The rest of the paper is organized as follows. We introduce the autonomous quantum dot heat engine model and its mathematically equivalent non-autonomous two-level model in Sec. II. First, the global optimization of power in the entire parameter space is presented in Sec. III. In Secs. IV and V, we present our main results for the optimization with various constraints and discuss its non-universal feature in power-maximizing efficiency. We conclude with the summary and a remark on future work in Sec. VI.

ii.1 Quantum dot heat engine model

We take a quantum dot heat engine introduced in Ref. Esposito2009EPL (), which is composed of a quantum dot whose energy level EG is controlled by the gate voltage where a single electron can occupy, in contact with two leads, denoted by R1 and R2 at different temperatures (T1>T2) and chemical potentials (μ1<μ2<EG), respectively, as shown in Fig. 1.
For notational convenience, we define the energy level of the quantum dot as EQD≡EG−μ1>0
and the chemical potential difference as Δμ=μ2−μ1>0.
Experimentally, it is possible to control EQD by tuning the gate voltage connected to the quantum dot and
Δμ by tuning the source-drain voltage connected to the leads Kouwenhoven1997 ().

The transition rates of the electron to the quantum dot from R1 and R2 are given as the following Arrhenius form,

q/~q=e−EQD/T1,

(1)

ϵ/~ϵ=e−(EQD−Δμ)/T2,

with q (ϵ) from R1 (R2) to the quantum dot and ~q (~ϵ) vice versa.
Here, we set the Boltzmann constant kB=1.
Denoting the probability of occupation in the quantum dot by Po and its complementary probability (of absence) by Pe=1−Po,
the probability vector |P⟩=(Po,Pe)T is described by the master equation

d|P⟩dt=(−~q−~ϵq+ϵ~q+~ϵ−q−ϵ)|P⟩.

(2)

For simplicity, tunneling rates between the quantum dot and the leads are chosen as q+~q=ϵ+~ϵ=1.
Generalization to arbitrary finite rates does not change our main conclusions. With these normalized rates,
we find the condition for parameters as 0≤ϵ,q≤1/2.
The steady-state solution is easily obtained as

Po,ss=12(q+ϵ),Pe,ss=12(2−q−ϵ),

(3)

with

EQD=T1ln[(1−q)/q],EQD−Δμ=T2ln[(1−ϵ)/ϵ].

(4)

The probability currents from R1 to the quantum dot and that from the quantum dot to R2 are then,

I1

=Pe,ssq−Po,ss(1−q)=12(q−ϵ),

(5)

I2

=Po,ss(1−ϵ)−Pe,ssϵ=12(q−ϵ),

respectively, and they are identical to each other, which represents the conservation of the particle flux. From now on, we denote
this steady-state particle flux carrying the energy current by

J≡12(q−ϵ).

(6)

The heat production rate to the quantum dot from R1 and that from the quantum dot to R2 are

˙Q1

=JEQD,

(7)

˙Q2

=J(EQD−Δμ).

A particle moving from the hot lead R1 to the cold lead R2 gains the energy Δμ, which can be used later
as work against an external device. Thus, the idealized power of the engine is defined as

˙W=˙Q1−˙Q2=JΔμ,

(8)

by the first law of thermodynamics. With Δμ>0, we need the condition of q≥ϵ (non-negative J) for a proper heat engine. It is clear that the tight coupling condition is satisfied in our model because the heat currents are proportional
to the work current with proportionality constants given by the non-vanishing ratio of energy control parameters.

The efficiency of the engine is given by the ratio

η=˙W˙Q1=ΔμEQD=1−T2ln[(1−ϵ)/ϵ]T1ln[(1−q)/q],

(9)

which is independent of temperatures and the particle flux J.
By adjusting temperatures to approach the limit of ϵ→q from below,
η can reach the maximum (Carnot) efficiency HuangBook (); Carnot1824 (),

ηC=1−T2T1.

(10)

The total entropy production rate in the steady state is given by the net entropy change rate of the leads;

˙S=−˙Q1T1+˙Q2T2≥0.

(11)

ii.2 Cyclic two-level heat engine model

The autonomous quantum dot heat engine introduced in Sec. II.1 is in fact equivalent to a simple non-autonomous
cyclic two-level heat engine described in Fig. 2.
The two-level system is characterized by two discrete energy states composed of the ground state (E=0) and the excited state (E=E1 or E=E2, depending on the contacting reservoir). The transition rates from the ground state to the excited state are denoted by q and ϵ, respectively, and their reverse processes by ~q and ~ϵ. We assume E1>E2 and T1>T2.

The system is attached to two different reservoirs: R1 with temperature T1 during time τ1, and R2 with temperature T2 during time τ2, and the adiabatic work extraction and insertion occur in between. Although the amount of energy unit involving the work exchange is the same (E1−E2), the net positive work is achievable due to the difference in the population of the excited states at the end of contact with R1 and R2, which is determined by model parameters. Then, the mathematical formulation is exactly the same as the quantum dot engine if we use the following mapping from the energy variables in the quantum dot engine in Sec. II.1:

E1≡EQD,E2≡EQD−Δμ.

(12)

Figure 2: Schematic illustration of a simple two-level heat engine, composed of two energy levels coupled with two heat reservoirs R1 and R2.

Using the same formalism as in the autonomous quantum dot engine except for the explicit time dependence, the net work per cycle
(cyclic period τ=τ1+τ2) in the cyclic steady state is given as

Wnet,two-level=(1−e−τ/2)2(q−ϵ)1−e−τ(E1−E2),

(13)

assuming τ1=τ2=τ/2 for simplicity. The τ-dependent factor (1−e−τ/2)2/(1−e−τ)
is decoupled from the rest of the formula and thus is just an overall factor. The decoupling holds regardless of the τ1=τ2 condition; the overall factor becomes (1−e−τ1)(1−e−τ2)/[1−e−(τ1+τ2)]. The mean power is then given by

Wtwo-levelτ=(1−e−τ/2)2(q−ϵ)τ(1−e−τ)(E1−E2),

(14)

which decreases monotonically with τ.

This result is exactly the same as the power for the quantum dot engine in
Eq. (8) by replacing the current J by Jcyc=a(τ)J with a(τ)=2(1−e−τ/2)2/[τ(1−e−τ)].
In fact, all formulas for various other quantities are also written with Jcyc instead of J, thus the analysis for the quantum dot engine
in the following sections should apply to the cyclic two-level heat engine with a trivial overall factor a(τ).

In this section, we investigate the efficiency at maximum power for the quantum dot engine.
We rewrite Eq. (8) for power in terms of q and ϵ as

˙W(q,ϵ)=12(q−ϵ)[T1ln(1−qq)−T2ln(1−ϵϵ)].

(15)

The condition for a proper heat engine with non-negative power (˙W≥0) further restricts the parameter space of (q,ϵ)
with

1−qq≤1−ϵϵ≤(1−qq)T1/T2.

(16)

Note that the lower bound corresponds to J=0 (the reversible limit with ˙S=0)
and the upper bound corresponds to Δμ=0 (no work limit with ˙W=0).

For given T1 and T2, the power can be maximized at (q∗,ϵ∗) inside the above restricted parameter space,
which satisfies

∂˙W∂q∣∣∣(q∗,ϵ∗)=∂˙W∂ϵ∣∣∣(q∗,ϵ∗)=0,

(17)

(see details in Appendix A).
The efficiency at maximum power, ηop, can be obtained from Eq. (9) with (q∗,ϵ∗), which
is a function of the temperature ratio T2/T1(=1−ηC). This function cannot be written in a closed form with ηC, but
its expansion near equilibrium (small ηC) is given by

ηop=12ηC+18η2C+7−24a0+24a2096(1−2a0)2η3C+O(η4C),

(18)

with a0=q∗|ηC=0+≈0.083222, which is the solution of
2/(1−2a0)=ln[(1−a0)/a0]. The same expression was reported previously in equivalent models Esposito2009EPL (); Toral2016 ().

Figure 3: The efficiency at maximum power ηop with respect to both q and ϵ versus the Carnot efficiency ηC. The CA efficiency ηCA in Eq. (19) and
and the ηC→1 asymptote in Eq. (21) are also shown.
The inset shows the magnified view of the region 0.97<ηC<1.

The two efficiencies share the same coefficients up to the quadratic terms in the expansion, which
are known to be universal due to tight-coupling between thermodynamic fluxes and the left-right
symmetry VanDenBroeck2005 (); Esposito2009PRL (). The third order coefficient (≃0.077492) in Eq. (18), however, is
different from 1/16 (=0.0625) for the ηCA.
Plots of ηop and ηCA against ηC
are shown in Fig. 3 for comparison.

The asymptotic behavior of ηop near ηC=1 is given by

ηop=1+(1−b0)(1−ηC)ln(1−ηC)+O[(1−ηC)],

(21)

with b0=q∗|ηC=1−≈0.217812 which is the solution of 1/(1−b0)=ln[(1−b0)/b0]. This result is also shown in Fig. 3 for comparison with ηCA.

For given T1 and T2, we fix the quantum dot energy and one of the chemical potential.
We vary Δμ (thus ϵ) with fixed EQD (so fixed q).

iv.1 efficiency at maximum power

For given q (or EQD), we find ϵ∗ maximizing the power in Eq. (15) with
∂˙W/∂ϵ|ϵ∗=0 in the parameter space restricted
by Eq. (16). A straightforward calculation similar to the global optimization in Sec. III
yields the efficiency at maximum power for small ηC as

ηop=12ηC+EQD16T2tanh(EQD2T2)η2C+O(η3C).

(22)

The linear coefficient 1/2 may be regarded as natural due to the tight-coupling condition VanDenBroeck2005 () in our model.
More detailed discussion on this 1/2 universality will be given later in Sec.V.2.

The quadratic coefficient is not universal, depending on the system parameter EQD, thus
differs in general from the universal value 1/8 representing the left-right symmetry.
This implies that the left-right symmetry should be considered not only in the engine device by itself,
but also in the allowed parameter space which is broken in this local optimization case.
We note that the universal value
1/8 is recovered for the special case of

EQDT2tanh(EQD2T2)=2.

(23)

Plots of ηop against ηC are shown in Fig. 4. It is interesting to note that
the asymptotic behavior of ηop near ηC=1 is quite different from that in the case of the global optimization (see Sec. III) and its leading order is given by
ηop≈1−α0+O(1−ηC)
with α0 satisfying the equation 1=α0+(T2/EQD)sinh[α0EQD/T2]. Note that
0≤α0≤1/2.

Figure 4: The efficiency at maximum power ηop versus ηC for EQD=1 and T2=1.
The black thick curve represents the exact one. The red thin line is drawn from the expansion
in Eq. (22) up to the quadratic order, which is very close to the exact one.
For comparison, we also plot the ηC/2+η2C/8 curve (purple thin line), which differs significantly.

iv.2 irreversible thermodynamics approach

Near equilibrium, it is useful to analyze a heat engine in the viewpoint of irreversible thermodynamics Groot (); SSheng2014 (); SSheng2015 ().
The total entropy production rate in Eq. (11) can be written as

˙S=˙Q1(1T2−1T1)−˙WT2≡JtXt+J1X1,

(24)

with
the thermal flux

Jt=˙Q1=JEQD,

(25)

the thermal force representing the temperature gradient

Xt=1T2−1T1=ηCT2,

(26)

the mechanical flux

J1=−JT2,

(27)

and the mechanical force representing the chemical potential gradient,

X1=ΔμT22.

(28)

Accordingly, the product of mechanical flux and mechanical force leads to the power

˙W=JΔμ=−T2J1X1.

(29)

The condition Xt=X1=0 corresponds to the thermal and mechanical equilibrium state with ˙S=˙W=0.

We expand the particle flux J in Eq. (6) for small forces Xt and X1 (small ηC and Δμ)
and find, after some algebra,

We optimize power in Eq. (29) with respect to X1 and find
the optimal X∗1 up to the quadratic order of Xt as

X∗1=−ξ2Xt+γξ28X2t.

(33)

Since the efficiency is given by

η=˙W˙Q1=−J1X1T2Jt=−X1T2ξ,

(34)

the efficiency at maximum power is obtained as

ηop=12ηC−ξγ8T2η2C+O(η3C),

(35)

which is obviously the same as that in Eq. (22). The condition of Eq. (23)
to get the universal quadratic coefficient 1/8 is equivalent to the “energy-matching condition” described in Ref. SSheng2015 ().

For given T1 and T2, we fix both chemical potentials and vary EQD to find the power maximum.
This situation is natural and easily realizable experimentally for a quantum dot engine where the source-drain voltage is fixed, while the gate voltage is adjusted to maximize the power Kouwenhoven1997 (); YSLiu2013 (); Humphrey2002 (); Jordan2013 (). It is in contrast to the previous cases where the maximum power is obtained by adjusting either or both of the source-drain voltages.

v.1 efficiency at maximum power

It is convenient to rewrite the expression for power in Eqs. (8) and (15) in terms of
energy variables Δμ and EQD as

˙W=12(e−EQD/T11+e−EQD/T1−e−EQD/T2eΔμ/T21+e−EQD/T2eΔμ/T2)Δμ.

(36)

For fixed Δμ>0, ˙W varies with EQD in the parameter range of EQD≥Δμ/ηC (q≥ϵ). Note that the boundary point ErQD=Δμ/ηC is a reversible one, where ˙W=˙S=0 along with
η=ηC.

As EQD increases from the reversible point, ˙W increases first but should decrease later after an optimal point because
Eq. (36) indicates that ˙W should vanish as EQD→∞.
The asymptotic point (EQD=∞) is special with the particle current J=0 in Eq.(6) but
q/ϵ=e−Δμ/T2≠1 (broken detailed balance).
The optimal point with maximum power is obtained by

∂˙W∂EQD∣∣∣EQD=E∗QD=0,

(37)

where the optimal E∗QD satisfies

e−E∗QD/T1(1+e−E∗QD/T1)2T2T1=e−E∗QD/T2eΔμ/T2(1+e−E∗QD/T2eΔμ/T2)2.

(38)

Figure 5: The efficiency at maximum power ηop for Δμ=1 and T2=1.
The black thick curve represents the exact one. It clearly shows that the slope for small ηC is 1, rather than 12.
The red thin line is drawn from the expansion in Eq. (40) up to the quadratic order, which is very close
to the exact one up to ηC≈0.4.

First, consider the asymptotic behavior near small ηC. The reversible point ErQD=Δμ/ηC diverges as well as
the optimal point E∗QD. Keeping the lowest order terms of e−E∗QD/T2 in Eq. (38),
we easily obtain

E∗QD=ΔμηC−T2ηCln(1−ηC).

(39)

Inserting this into Eq. (9),
we finally arrive at the efficiency at maximum power as

ηop=ηC−T2Δμη2C+O(η3C).

(40)

In contrast to the previous cases, the linear coefficient in the expansion deviates from the 1/2 universality and becomes unity along with the negative quadratic coefficient. This example clearly illustrates that this seemingly robust universality for conventional tight-coupling engines VanDenBroeck2005 () can be also violated, depending on the type of restricted control-parameter spaces used in the power
maximization. In the next subsection, we will discuss about the violation of the 1/2 universality in the perspective of irreversible thermodynamics
and the singular behaviors of thermodynamic and mechanical fluxes.

Next, we consider near ηC≈1. In Fig. 5 where the exact result (numerically obtained) is displayed for all values of ηC, we note that ηop does not increase monotonically with ηC and
vanishes at ηC=1 with a singularity. After some algebra, we find indeed a logarithmic singularity
such as ηop≈(Δμ/T2)/[−ln(1−ηC)].

v.2 irreversible thermodynamics approach

As EQD is varied with fixed Δμ, the mechanical force
X1 in Eq. (28)
cannot be used as a mechanical force variable. Instead, we take the mechanical force defined as

X2=1EQD

(41)

and the corresponding mechanical flux J2 should be given as

J2=−˙WT2X2=−JΔμT2EQD.

(42)

Then, we write the entropy production rate in the standard form as

˙S=JtXt+J2X2,

(43)

with the same thermal flux Jt=JEQD and force Xt=ηC/T2
in Eqs. (25) and (26).

In contrast to the previous case with fixed EQD in Sec. IV,
the condition Xt=X2=0 does not correspond to equilibrium
because of non-zero Δμ.
At the X2=0(EQD=∞) point, the particle current J vanishes (exponentially) as well as ˙Q1=˙W=0
with η=Δμ/EQD=0 (see Eq. (9)). As mentioned in the previous subsection, the detailed balance is broken due to
q/ϵ≠1. Even though ˙S=0 at this point, the average entropy production per
one particle transfer diverges as ˙S/J≈EQDηC/T2, which reveals its irreversible feature.
(A similar situation was discussed in JSLee2017 ().)
Therefore, although our approach is dealing with vanishing fluxes in the limit of Xt→0 and X2→0,
it is not technically the conventional irreversible thermodynamics used in Refs. Groot (); SSheng2014 (); SSheng2015 (),
which is a perturbation theory based on the true equilibrium state.
Nevertheless, in the following, we present the same type of irreversible thermodynamics analysis and
its implication for better understanding of the situation.

(a)

(b)

Figure 6: Comparison between (a) the fixed-EQD case and (b) the fixed-Δμ case, in terms of the mechanical flux (we plot the negative value of the flux for better visualization) and power,
where we set T1=1 and T2=1/2.
In (a), we plot J1 and ˙W against the mechanical force X1 at EQD=1. Both J1 and ˙W vanish at X1=−ξXt=EQDηC/T22=2, but only ˙W vanishes at X1=0. In (b), we plot J2 and ˙W against X2 at Δμ=1. In this case, both J2 and ˙W vanish at both X2=−ξ′Xt=ηC/Δμ=1/2 and X2=0.
For each case, we indicate the optimal values of X∗1 and X∗2 at maximum power.

Let us start with the tight-coupling condition between Jt and J2,

Jt/J2=−T2/Δμ≡ξ′,

(44)

which is a constant in the optimization process in this section.
This condition guarantees that the reversible condition ˙S=0 can be achieved at non-zero forces with X2=−ξ′Xt,
similar to the standard irreversible thermodynamics discussed in Sec. IV.2.
Expansion of the mechanical flux in Eq. (42) for small forces Xt and X2 leads to

J2=Δμ2T2X2e−1T2X2(eΔμ/T2−eXt/X2)+O(e−2T2X2,e2(T2Xt−1)T2X2),

(45)

which vanishes as X2∼Xt→0 with an essential singularity rather than linearly seen in Sec. IV.2. This implies that the linear irreversible thermodynamic
analysis is not applicable to our case.

Plots of the power ˙W and the mechanical flux J1 for the fixed-EQD case in Sec.IV.1 and J2 for the fixed-Δμ case in this section are shown for comparison in Fig. 6.
For the fixed-EQD case shown in Fig. 6(a), ˙W should be approximated as a simple parabola for very small ηC
(thus very small parameter interval), because the limiting behaviors near both boundaries (X1=0 and X1=−ξXt) are linear,
which is usually the case in most optimization procedures.
Then the optimal X∗1 is right at the middle point (X∗1=−ξXt/2).
On the other hand, the efficiency increases linearly such as η=Δμ/EQD∼X1 and reaches ηC at the
reversible point (X1=−ξXt). Thus
we can easily expect the 1/2 universality (ηop≃ηC/2) at maximum power, in general.

However, for the fixed-Δμ case, the functional behavior of ˙W near X2=0 is anomalous with an essential
singularity, seen in Eq. (45) and in Fig. 6(b). When the parameter interval becomes very small (small ηC),
one can easily expect the optimal X∗2 should approach the reversible point X2=−ξ′Xt, leading to
ηop≃ηC found in Eq. (40).

For simple analysis, we consider a nonlinear leading term of an arbitrary order in the mechanical flux as

J2=L′(X2+ξ′Xt)Xn2,

(46)

which vanishes at the reversible point X2=−ξ′Xt and also at X2=Xt=0. We optimize the power ˙W=−T2J2X2 in Eq. (42) with respect to
X2 and find the optimal X∗2 by

X∗2=−n+1n+2ξ′Xt,

(47)

and the efficiency at maximum power is obtained as

ηop=n+1n+2ηC.

(48)

The linear case (n=0) yields ηop=ηC/2 for the tight-coupling heat engine VanDenBroeck2005 ()
as expected.
However, our case with an essential singularity in Eq. (45) corresponds to the n→∞ limit,
leading to ηop≃ηC, which is consistent with our result in Eq. (40), up to the leading order. We remark that our heat engine provides only three possible values of the linear coefficient as 1/2, 1, and 0
(varying EQD and Δμ together such as ηCEQD=Δμ+bT2 with b>0).

Figure 7: Comparison between local and global optimizations.
(a) The efficiency at maximum power and (b) the maximum power in local optimizations scaled by those in global optimizations
with T2=1. The purple and green curves correspond to the fixed-EQD case with
EQD=2 and 2.4. The red and black curves correspond to the fixed-Δμ case with
Δμ=1 and 1.5.

The effectiveness of an engine should be featured by a high efficiency and a high power output.
However, there is a trade-off relation between the power and the efficiency Shiraishi2016 (), which
does not allow both merits simultaneously. In previous subsections, we show that, for small ηC (more realistic situations),
the power optimization with fixed Δμ provides us a higher efficiency at maximum power than that in the global optimization discussed in Sec. III. But it is also obvious that its power output cannot be larger than that at the global maximum.

The efficiencies at maximum power ηop
in two local optimizations are shown in Fig. 7(a) in comparison with that
in the global optimization. As expected, ηop for the fixed-Δμ case is larger than that for the
global optimization for a rather wide range of ηC (ηC≲0.5).
We also plot the maximum power in local optimizations scaled by the global optimum value
Fig. 7(b). We note that the maximum power for the fixed-Δμ case
reaches up to a significant fraction of the global optimum value.
For example, the case of Δμ=1 at ηC≃0.3 gives about 30% larger ηop than that for the global optimization case and reaches about 70% of the global maximum power exp1 (). This engine at these parameter values may be viewed
as “more effective” than the globally optimized engine in some specific situations preferring a good efficiency.

We have demonstrated that a quantum dot heat engine exhibits various nonuniversal forms of the efficiency at maximum power ηop.
In particular, compared to the global or local optimization with varying source-drain voltages, the single-parameter optimization by controlling the gate voltage of the quantum dot for fixed source-drain voltages
reveals ηop≈ηC for small ηC, which breaks the so-called 1/2 universality
(ηop≈12ηC). This universality has been believed to be robust for any engine with the tight-coupling condition of thermodynamics fluxes.

We have investigated the origin of this universality break down in terms of irreversible thermodynamics and
a singular behavior of the mechanical current. In fact, the absence of linear response regime of thermodynamic fluxes may yield
various values of the linear coefficient in the standpoint of irreversible thermodynamics.
Our case turns out to be an extreme case with an essential singularity in the mechanical current, which
makes the efficiency at maximum power close to the Carnot efficiency.
A recent experimental study for a quantum dot system exp () shows results consistent with our theoretical finding.

The two mathematically identical two-level heat engine models (autonomous engine and non-autonomous cyclic engine) introduced in Sec. II would naturally involve quantum effects in reality when we take atomic-scale systems. A direction for future works would be taking into account the genuine quantum effects Scovil1959 (); Uzdin2015 (); KUP (). It would be also interesting to study the equivalence of the autonomous and non-autonomous models at the quantum level KUP ().

Acknowledgements.

We thank Hyun-Myung Chun, Jae Dong Noh, Hee Joon Jeon, and Sang Wook Kim for fruitful discussions and comments.
This research was supported by the NRF Grant No. NRF-2017R1D1A1B03030872 (JU) and
2017R1D1A1B06035497 (HP), and by the Gyeongnam National University of Science and Technology Grant in 2018–2019 (SHL).

Appendix A Global optimization

By eliminating the left-hand side of Eqs. (49a) and (49b), we obtain
the following simple relation

T2q∗(1−q∗)T1ϵ∗(1−ϵ∗)=1,

(50a)

or

ϵ∗=12[1−U(ηC,q∗)],

(50b)

with

U(ηC,q∗)≡√4ηCq∗(1−q∗)+(1−2q∗)2.

(51)

By substituting ϵ∗ as a function of q∗ in Eq. (50b) to Eq. (49a) or Eq. (49b),
we get the optimum condition

ln(1−q∗q∗)−T2T1ln[1+U(ηC,q∗)1−U(ηC,q∗)]=q∗−12+12U(ηC,q∗)q∗(1−q∗).

(52)

Furthermore, the condition in Eq. (52) leads to the following form of ηop from Eq. (9),

ηop=q∗−12+12U(ηC,q∗)q∗(1−q∗)ln[(1−q∗)/q∗].

(53)

In order to calculate the efficiency at maximum power for given T2/T1, first find the q∗ value satisfying Eq. (52) and substitute the q∗ value to Eq. (53).
As Eq. (52) is a transcendental equation, the closed-form solution for ηop is not possible in general.

We study analytically asymptotic behaviors of ηop near ηC=0 and ηC=1.
First, examine the case for small ηC, using the series expansion of q∗ with respect to ηC as

(54)

Substituting Eq. (54) into Eq. (52) and expanding the equation with respect to ηC again, we obtain

c1ηC+c2η2C+c3η3C+O(η4C)=0,

(55)

where cn is a function of a set of coefficients {a0,⋯,an−1}.
To satisfy Eq. (55),
each cn should be identically zero. From c1=0, we can easily find

21−2a0=ln(1−a0a0),

(56)

from which we get a0≈0.083222. This serves as the lower bound of q∗.
From c2=0 and c3=0, we can express a1 and a2 in terms of a0. From Eq. (50b),
we can also find ϵ∗ as ϵ∗=q∗−[a0(1−a0)/(1−2a0)]ηC+⋯.

With the relations of coefficients in hand, we find the asymptotic behavior of ηop in Eq. (53) by expanding it with respect to ηC after substituting q∗ as the series expansion of ηC in Eq. (54).
Then, we obtain the expression in Eq. (18) in the main text,

ηop=12ηC+18η2C+7−24a0+24a2096(1−2a0)2η3C+O(η4C).

(57)

With this method, we are able to find the coefficients in terms of a0 up to an arbitrary order in principle.

For ηC≃1, we need to take into account a logarithmic singularity,
arising from ln[1−U(ηC,q∗)]∼ln(1−ηC) in Eq. (52).
We take a singular series expansion of q∗ with respect to 1−ηC as

q∗=b0+b′1(1−ηC)ln(1−ηC)+b1(1−ηC)+O[(1−ηC)2].

(58)

Substituting Eq. (58) into Eq. (52) and expanding the equation with respect to
1−ηC, we can identify the equation for b0 as

11−b0=ln(1−b0b0),

(59)

from which we get b0≈0.217812. This serves as the upper bound of q∗.
We also find b′1=b0(1−b0)2 and b1=b0(1−b0)2(1+ln[b0(1−b0)]).
Putting all these together into Eq. (53), we obtain

A similar result was reported experimentally in the lower right panel of Fig. 3(c) in exp (), where, with ηC≃0.36,
ηop is about 25% larger than ηCA and the power output is about 65% of its global maximum.